Simulation driven reduced order modelling is gaining traction in the scientific computing community [CSGRR22], both when dealing with parameter dependent families of problems as well as in the cases when operator functions need to be approximated. In this context one is also considering deep learning models which describe the dependence of the result of a simulation on parameters of the problem. As prototypes for our study we will consider applications of operator functions and functional calculus in control theory, e.g. [GLNT22], as well as in the context of approximating solutions of the differential equations with nonlocal terms, e.g. [EGG22]. In both of these instances the numerical approximation was constructed from samples of the appropriate resolvent function taken at different points in the resolvent set and evaluated at randomly constructed elements of the state space. Evaluating the resolvent function at a given point in the resolvent set and applying it on an element of the state space, amounts to solving a Helmholtz-like problem. A typical task in data driven modelling is to reconstruct the system (resolvent) given only these solutions, which can be seen as observations of a numerical experiment. Such scenaria arise when the available implemented method for solving the Helmholtz problem cannot be altered, but a reduced order model of the systems needs to be built for further optimization or computation. This is what we mean by the simulation driven approach to reduced order modelling. We will explore this concept in the specific setting of more general reduced order (surrogate) modelling of a resolvent function associated both with linear as well as nonlinear operator formulations. Further, we will consider other approaches to sampling the resolvent (solving the Helmholtz like problems) such as those which involve boundary integral formulations. Boundary integral formulations reduce the problem posed in an infinite domain to the problem for a compact operator acting on a trace space of a compact manifold and so allow for treatment of the original problem by the use of appropriate finite element discretizations. Non-local operators are typically discretized by dense matrices. Subsequently, the use of low rank or stochastic approximations of these operators, e.g. [NT22], is necessary to be able to devise efficient computational methods as well as to regularise solutions by removing discretization artefacts (e.g. by truncating discrete approximate modes which are not resolving features of the continuous model due to the coarseness of the discretization, cf. [GH22]). A central notion when approaching the construction of efficient numerical methods is the use of functional calculus based on numerical contour integration [GG15], [JSSGS22]. In order to achieve numerical efficiency, these methods need to be boosted by randomised low dimensional surrogate modelling of the resolvent function [MOBG22], see Figure 3.

In this work package we will combine the techniques from [NT22] and [MOBG22] together with modern multi mesh discretization techniques [GS21] to construct adaptive and efficient methods for approximating the solutions of partial differential equations with nonlocal components such as memory coefficients or fractional time derivatives used to describe anomalous diffusion or wave like propagation. Also, we will consider resonance models where boundary element discretizations are used to numerically treat problems posed in infinite domains. We will further improve the efficiency and the scope of application of the algorithm from [GLNT22] to include more general initial state control problems of the same type.

Literature:

- [CSGRR22] Cuomo, S.; Di Cola, V.S.; Giampaolo, F. et al.:
*Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next.*J Sci Comput 92, 88 (2022). - [GLNT22] Grubišić, L; Lazar, M; Nakić, I; Tautenhahn, M:
*Optimal control of parabolic equations – a spectral calculus based approach*. Submitted to SIAM Journal of control and optimization, pp. 25 (2022) - [EGG22] Engstrom, C.; Giani, S.; Grubišić, L:
*A spectral projection based method for the numerical solution of wave equations with memory*. Applied Mathematics Letters**127**, 107844 (2022) - [NT22] Nakatsukasa, Y.; Tropp, J.:
*Fast & Accurate Randomized Algorithms for Linear Systems and Eigenvalue Problems*, ArXiv arXiv:2111.00113v2 (2021). - [GG15] Grubišić, L.; Grbić, A.: Discrete perturbation estimates for eigenpairs of Fredholm operator-valued functions. Applied Mathematics and Computation 267, 632-647 (2015)
- [JSSGS22] Jorkowski, P.; Schmidt, K.; Schenker, C.; Grubišić, L.;Schuhmann, R.: Adapted contour integration for nonlinear eigenvalue problems in wave-guide coupled resonators. IEEE Transactions on Antennas and Propagation
**70**(1), 499 – 513 (2022) - [MOBG22] Mensah, GA; Orchini, A; Buschmann, PE; Grubišić, L: A subspace-accelerated method for solving nonlinear thermoacoustic eigenvalue problems. Journal of Sound and Vibration
**520**, 116553 (2022) - [GS21] Giani, S.; Solin, P.: Solving elliptic eigenproblems with adaptive multimesh hp-FEM, Journal of Computational and Applied Mathematics, Volume 394, 113528, (2021)
- [GH22] Grubišić, L.; Hakula, H.: High order approximations of the operator Lyapunov equation have low rank. BIT Numerical Mathematics, 1-27, https://doi.org/10.1007/s10543-022-00917-z, (2022)